Let $E$ be a homogeneous subset of $\mathbb{R}$ in the
sense of Carleson. Let $\mu$ be a finite positive measure on
$\mathbb{R}$ and $H_\mu(x)$ its Hilbert transform. We prove that
$\lim_{t\to\infty} t|E\cap\{x : |H_\mu(x)|>t\}|=0$ if and only if
$\mu_s(E)=0$, where $\mu_s$ is the singular part of $\mu$.

In this expository talk we relate the spectral properties of a discrete
Schr"odinger operator on a d-dimentional lattice to its dynamical
features. Dynamical quantities of interest include Fourier transforms of
spectral measures, time averaged moments of the position operator, as well
as time-averaged observables for a compact operator. The RAGE theorem in
its various formulations predicts the asymptotic behaviour of these
quantities for any state in the continuous subspace of the Hilbert space:
observables for a compact operator decay to zero, whereas the moments of
the position operator asymptotically diverge. In order to quantify this
decay/divergence, we present a decomposition of the spectral measure with
respect to Hausdorff measures of dimension $\alpha \in [0,1]$. This
decomposition due to Rogers and Taylor generalizes the classical
decomposition of the spectral measure w.r.t. Lebesgue measure into pure
point and continous component. Whereas for $\alpha = 1$ it recasts the
classical result, for $\alpha < 1$ one obtains a decomposition different
to the classical one. For each Hausdorff dimension, the spectral measure
then splits in an $\alpha$-continuous and an $\alpha$-singular component.
$\alpha$-continuous measures are shown to be limits of uniformly $\alpha$
H"older continuous (U$\alpha$H) measures w.r.t. to a suitable topology.
For U$\alpha$H spectral measures lower and upper bounds for various
dynamical quantities are available.

references:
Y. Last, Quantum dynamics and decompositions of singular continuous
spectra, J. Funct. Anal 142, 406-445 (1996).
W. Kirsch: An invitation to random Schr"odinger operators,
arXiv:0709.3707v1[math-ph].
G. Teschl: Mathematical Methods in Quantum mechanics with application to
Schroedinger operators, Graduate Studies in Mathematics, Amer. Math. Soc.,
Providence, 2008. (to appear).

Reflectionless measures are interesting objects because
they arise as limiting measures of spectral measures of arbitrary
Schr"odinger operators with some absolutely continuous spectrum.
In this talk, I'd like to review the definition and some background
material and then discuss more recent work, joint with Alexei Poltoratski,
on reflectionless measures.